Periods of the motivic fundamental groupoid of $\boldsymbol{\mathbb{P}^{1} \diagdown \lbrace 0, μ_{N}, \infty \rbrace}$ (1603.05155v1)

Abstract: In this thesis, following F. Brown's point of view, we look at the Hopf algebra structure of motivic cyclotomic multiple zeta values, which are motivic periods of the fundamental groupoid of $\mathbb{P}^{1} \diagdown \lbrace 0, \mu_{N}, \infty \rbrace $. By application of a surjective \textit{period} map (which, under Grothendieck's period conjecture, is an isomorphism), we deduce results (such as generating families, identities, etc.) on cyclotomic multiple zeta values, which are complex numbers. The coaction of this Hopf algebra (explicitly given by a combinatorial formula from A. Goncharov and F. Brown's works) is the dual of the action of a so-called \textit{motivic} Galois group on these specific motivic periods. This entire study was actually motivated by the hope of a Galois theory for periods, which should extend the usual Galois theory for algebraic numbers.